Chapter 2 Notes

1. Intro – In this section we will learn about non-real numbers. The motivation for this is to find the solutions to equations that we say “____________________” Chapter 2 Notes Section 2.4 Complex Numbers

2. 2.4 Notes (continued) • The imaginary unit (i) – • ______________________________________ • An imaginary number is the imaginary unit with a coefficient. Like, __________________ • The imaginary numbers combine with real numbers to make up a new set of numbers called ___________________

3. 2.4 Notes (continued) • For example

4. 2.4 Notes (continued) • Complex numbers – If a and b are real numbers, the number a + bi is a complex number, and it is said to be written in ______________________ • If ________, the number _________, is a real number. • If ________, the number _________is called an imaginary number. • A number in the form bi, where b ≠ 0, is called a ____________________.

5. 2.4 Notes (continued) • Operations with complex numbers • Let there exist two complex numbers (a + bi and c + di) • Addition • Subtraction • Multiplication • Division – can not be done so simply, we must about complex conjugates first.

6. 2.4 Notes (continued) • Perform the following operations

7. 2.4 Notes (continued) • Complex conjugates are two complex numbers that, when multiplied, __________________ • For any complex number _________, it’s complex conjugate will be _________ • For example

8. 2.4 Notes (continued) • Complex conjugates are important for ____________________________. • When dividing by a complex number, we must ________________________ • So we multiply top and bottom by the ___________________of the denominator

9. 2.4 Notes (continued) • Divide

10. 2.4 Notes (continued) • Divide

11. Examples

12. Examples